tldr: There is a temporary access trail that gets you around the washout! From where you can park your car to the hot springs is 10.9 miles. Bikes or wheeled vehicles are allowed on the road, but not the temporary access trail or the hot springs trail.
I left at about 11:45am from the parking lot where you can drive up to and didn't get to the hot springs until nightfall. Sign on the closed gate says it is 10.9 miles from that location to the hot springs, but you can use wheel vehicles if you like on the road. Someone had a fat tire bike and there were some bike trails in the snow going up to the trailhead. If you do that, there are a lot of logs down across the road and you would have to lift the bike over some of them and technically carry your bike on the bypass trail or try to ford the washout with it. Eek! I don't know how they got past the washout!
I just parked my car there and walked the mostly two lane paved road to the trailhead. I dropped my envelope at the Elwha Ranger Station along the way to the Hot Springs. The ranger at the Port Angeles Wilderness Information Center didn't even know if there would be envelopes up there!
Before the washout they have now built a new trail that is well marked on the way out and was a little tricky on the way back at night for me. Follow the yellow ribbons. I don't know how I got lost, but I follow the pink and white flags accidentally on the way back and had to scramble up an eroding cliff to get back up to the trail! (I have included a picture of the new trail that bypasses the washout.)
There is snow from the official trail head up to the hot springs anywhere from about 1" to 12" in depth that compacted from about half way to three quarters of the way up my gaiters. The log bridges are slippery, but they have handrails. The hot springs were lukewarm or even cool at some points probably due to the heavy rains. There is a huge fallen tree before one of the hot springs that I fell trying to get around and then just crawled under it on the way back.
I got so wet (it was raining, misting, or snowing the whole time) I ended up doing a 22 mile hike instead of an overnight because I needed to keep moving to stay warm. Fun and adventurous if you like that sort of thing! Bring the ten essentials! Hope you find this report helpful!